Final answer:
The lowest score for a grade A is approximately 83.225 and the highest score for a grade D is approximately 71.63.
Step-by-step explanation:
To find the lowest score for a grade A, we need to determine the cutoff point for the top 5% of students. Since the exam scores are approximately normally distributed with a mean of 75 and a standard deviation of 5, we can use the z-score formula.
The z-score formula is: z = (x - μ) / σ
Substituting the values, we have: z = (x - 75) / 5
We need to find the z-score corresponding to the top 5% of students, which is a z-score of 1.645. Solving the equation, we get: 1.645 = (x - 75) / 5. Rearranging the equation, we find x = 75 + (1.645 * 5). Therefore, the lowest score for a grade A is approximately 83.225.
To find the highest score for a grade D, we need to determine the cutoff point for the bottom 25% of students. Using the same z-score formula, we find the z-score corresponding to the bottom 25% is approximately -0.674. Solving the equation, we get: -0.674 = (x - 75) / 5.
Rearranging the equation, we find x = 75 + (-0.674 * 5). Therefore, the highest score for a grade D is approximately 71.63.